CHAPTER VII

SPACES OF CONTINUOUS FUNCTIONS

Spaces of continuous functions are second only to Hilbert spaces as
to their importance in functional analysis. Their definition makes it possible
to give a much more intuitive meaning to the classical notion of uniform
convergence. The most important results of the chapter are: 1. the Stone-
Weierstrass approximation theorem (7.3.1), which is a very powerful tool
for the proof of general results on continuous functions, by the device which
consists in proving these results first for functions of a special type, and then
extending them to all continuous functions by a density argument; 2. the
Ascoli theorem (7.5.7), which lies at the root of most proofs of compactness
in function spaces, and, together with (7.5.6), gives the motivation for the
introduction of the concept of equicontinuity.

The last section of Chapter VII introduces, as a useful technical tool in
the development of calculus, a category of functions which are classically
described as "functions with discontinuities of the first kind"; in an effort
towards a more concise expression, and to avoid one more use of the over-
worked term "regular," the author has tentatively introduced the neologism
"regulated functions" (corresponding to the French "fonctions reglees"),
which he hopes will not sound too barbaric to English-speaking readers.

1. SPACES OF BOUNDED FUNCTIONS

Let A be any set, F a real (resp. complex) normed space; a mapping
/of A into F is bounded if f (A) is bounded in F, or equivalently if sup ||/(0il

teA

is finite. The set #F(A) of all bounded mappings of A into F is a real (resp.
complex) vector space, since \\f(t) + g(t)\\ < ||/(f)|| + 110(011- Moreover, on
this space,

132a finite sequence of points in a prehilbert space E. The Gram determinant